This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.

1. Prime and Composite
Numbers
Greatest Common Divisor &
Least Common Multiple

2. A Prime Number has exactly two distinct positive
divisors.
A Prime Number can only be divided by itself and 1.
Prime Numbers:
2
3
A composite number has factors
5
other than itself and 1.
7
11
Example: 6 is composite because
13
its factors are 1, 2, 3 and 6.
17
19
23
29…

3. A factor tree can be used to find the prime factors
of a composite number.
24
24
8 3
6 4
2 4
2 3 2 2
2 2
Prime factorization Fundamental Theorem of
of 24: Arithmetic:
24 = 2 · 2 · 2 · 3 Each composite number can be
or written as a product of primes in
24 = 23 · 3 one way only.

4. Is 43 prime or composite?
If 43 is composite, it has prime factors
Divide 43 by prime numbers to see if they are divisors
43 2 = 21 R 1
43 3 = 14 R 1
43 5 = 8 R 3
43 7 = 6 R 1
43 11 = 3 R 10 How far did we
really have to go
43 is prime. before we could
determine that 43 is
prime?

5. What is the largest possible prime factor of 43? 5
Prime Square
2 4 43 2 = 21 R 1
3 9 43 3 = 14 R 1
5 25 43 5= 8R3
43
7 49 43 7= 6R1
11 121 43 11 = 3 R 10
13 169
17 289

6. Is 113 prime or composite?
Prime Square
2 4 Is 2 a divisor of 113? No
3 9 Is 3 a divisor of 113? No
5 25 Is 5 a divisor of 113? No
7 49 Is 7 a divisor of 113? No
113
11 121
13 169
17 289 113 is prime

7. Divisors of Composite Numbers
List all divisors of 12 1, 2, 3, 4, 6, 12
How many divisors does 1500 have?
1500
1500 = 22 · 31 · 53
15 100
Number of divisors:
3 5 10 10 (2 + 1)(1 + 1)(3 + 1)
= (3)(2)(4)
2 5 2 5 = 24

8. The Greatest Common Divisor (GCD) of two or more
integers is the largest integer that divides the numbers.
Find the GCD of 8 and 20
Intersection of sets method
Factors of 8: 1, 2, 4, 8
Factors of 20: 1, 2, 4, 5, 10, 20
Common Factors: 1, 2, 4
GCD(8, 20) = 4

9. Find the GCD of 8 and 20
Prime factorization method 8 20
8=2·2·2 4 2 4 5
20 = 2 · 2 · 5
2 2 2 2
2 2
GCD(8, 20) = 2 · 2 = 4

10. Find the GCD of 42 and 63
Intersection of sets method
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42,
Factors of 63: 1, 3, 7, 9, 21, 63
Common Factors: 1, 3, 7, 21
GCD(42, 63) = 21

11. Find the GCD of 42 and 63
Prime factorization method 42 63
42 = 2 · 3 · 7 6 7 9 7
63 = 3 · 3 · 7
2 3 3 3
3 7
GCD(42, 63) = 3 · 7 = 21

12. Euclidean Algorithm
The GCD(a, b) = GCD(r, b) where r is the remainder
when a is divided by b.
GCD(42, 63)
1 2
42 63 21 42
42 42
21 0 0 remainder
GCD = 21

13. GCD(1824, 7448)
4 12
1824 7448 152 1824
7296 152
152 304
304
0 0 remainder
GCD = 152
Check: Is 152 a the GCD of 1824 and 7448?
1824 ÷ 152 = 12
7448 ÷ 152 = 49

14. The Least Common Multiple (LCM) of two or more
numbers is the smallest number that is a multiple of
all the numbers.
Find LCM(8, 20)
Intersection of sets method:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …
Multiples of 20: 20, 40, 60, 80, …
Common Multiples: 40, 80, 120, …
LCM: 40

15. Find LCM(8, 20) 8 20
Prime factorization method 4 4 5
2
8=2·2·2 2 2
2 2
20 = 2 · 2 · 5
L C M
2 2 2 5 E O U
F U L
T N T
LCM(8, 20) = 2 · 2 · 2 · 5 = 40
O T I
V with P
E L
R E
S S